In this section we prove Theorem REF  using scattering theory. Such an approach was developed in [1]} for a fixed value of \(a\) , so we extend it here to an \(a\) -dependent framework. We first define the scattering amplitude \(S\)  and establish its main properties. Then we relate the number of edge modes with the winding number of \(S\)  along some limiting loop approaching \(\mathcal {C}_R\) . Furthermore, the Chern number \(C_+\)  is also related to a winding number of \(S\)  along some other loop, which is independent of \(a\) . Finally we show that the two winding numbers coincide.
